解SDD系统的最优性逼近
Approaching optimality for solving SDD systems
摘要：
我们提出了一个算法，在输入的$n$ -顶点$m$ -边加权图$G$和一个值$k$，产生一个{ \em增量稀疏} $\hat{G}$与$n-1 + m/k$边，使$G$与$\hat{G}$的条件数是有界以上的$\tilde{O}(k\log^2 n)$，概率$1-p$。该算法在时间上运行$$\tilde{O}((m \log{n} + n\log^2{n})\log(1/p)).$$因此，我们得到了一个算法，该算法在具有$m$非零元素的$n\times n$对称对角占优矩阵$A$和向量$b$的输入下，计算满足$||{x}-A^{+}b||_A<ε||A^{+}b||_A $的向量${x}$，在预期时间内$$\tilde{O}(m\log^2{n}\log(1/ε)).$$求解器基于增量稀疏器的重复应用，该增量稀疏器产生一个图链，然后将其用作递归预处理Chebyshev迭代
英文摘要：
We present an algorithm that on input of an $n$-vertex $m$-edge weighted graph $G$ and a value $k$, produces an {\em incremental sparsifier} $\hat{G}$ with $n-1 + m/k$ edges, such that the condition number of $G$ with $\hat{G}$ is bounded above by $\tilde{O}(k\log^2 n)$, with probability $1-p$. The algorithm runs in time $$\tilde{O}((m \log{n} + n\log^2{n})\log(1/p)).$$ As a result, we obtain an algorithm that on input of an $n\times n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector $b$, computes a vector ${x}$ satisfying $||{x}-A^{+}b||_A<ε||A^{+}b||_A $, in expected time $$\tilde{O}(m\log^2{n}\log(1/ε)).$$ The solver is based on repeated applications of the incremental sparsifier that produces a chain of graphs which is then used as input to a recursive preconditioned Chebyshev iteration.
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